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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemeiota | Structured version Visualization version GIF version |
Description: A translation is uniquely determined by one of its values. (Contributed by NM, 18-Apr-2013.) |
Ref | Expression |
---|---|
cdlemg1c.l | ⊢ ≤ = (le‘𝐾) |
cdlemg1c.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemg1c.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemg1c.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
cdlemeiota | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2780 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → (𝐹‘𝑃) = (𝐹‘𝑃)) | |
2 | simp3 1118 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝑇) | |
3 | cdlemg1c.l | . . . . . . 7 ⊢ ≤ = (le‘𝐾) | |
4 | cdlemg1c.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | cdlemg1c.h | . . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | cdlemg1c.t | . . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
7 | 3, 4, 5, 6 | ltrnel 36717 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
8 | 7 | 3com23 1106 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
9 | 3, 4, 5, 6 | cdleme 37138 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) → ∃!𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃)) |
10 | 8, 9 | syld3an3 1389 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → ∃!𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃)) |
11 | fveq1 6498 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑃) = (𝐹‘𝑃)) | |
12 | 11 | eqeq1d 2781 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑃) = (𝐹‘𝑃) ↔ (𝐹‘𝑃) = (𝐹‘𝑃))) |
13 | 12 | riota2 6959 | . . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ ∃!𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃)) → ((𝐹‘𝑃) = (𝐹‘𝑃) ↔ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃)) = 𝐹)) |
14 | 2, 10, 13 | syl2anc 576 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) = (𝐹‘𝑃) ↔ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃)) = 𝐹)) |
15 | 1, 14 | mpbid 224 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃)) = 𝐹) |
16 | 15 | eqcomd 2785 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝐹‘𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ∃!wreu 3091 class class class wbr 4929 ‘cfv 6188 ℩crio 6936 lecple 16428 Atomscatm 35841 HLchlt 35928 LHypclh 36562 LTrncltrn 36679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-riotaBAD 35531 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-1st 7501 df-2nd 7502 df-undef 7742 df-map 8208 df-proset 17396 df-poset 17414 df-plt 17426 df-lub 17442 df-glb 17443 df-join 17444 df-meet 17445 df-p0 17507 df-p1 17508 df-lat 17514 df-clat 17576 df-oposet 35754 df-ol 35756 df-oml 35757 df-covers 35844 df-ats 35845 df-atl 35876 df-cvlat 35900 df-hlat 35929 df-llines 36076 df-lplanes 36077 df-lvols 36078 df-lines 36079 df-psubsp 36081 df-pmap 36082 df-padd 36374 df-lhyp 36566 df-laut 36567 df-ldil 36682 df-ltrn 36683 df-trl 36737 |
This theorem is referenced by: cdlemg1cN 37165 cdlemg1cex 37166 cdlemm10N 37696 |
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